with the partial differential equation if for any smooth function o(t, r) P-Bk→0ask,h→0, the convergence being pointwise convergence at each grid point Example The laz- Friedrichs scheme对 Lax-Friedrichs格式: Pk.hg φm+1-是(mn+1+m-1),m 用 Taylor公式 omn+1=m±hpx+h2x±h3bx+O(h2) 我们可以得到 Pk,ho=ot+aor+okott-okh'oxr -ah'oxrr +O(h+k h+k) 这样当h,k→0,且k-1h2→0时,Pn-P→0. Definition. 4 A finite difference scheme Pk hUm=0 for a first-order equa tion is stable if there is an integer J and positive numbers ho and ko such that for any positive time T, there is a constant CT such that h∑m2≤cr∑∑mP2 jor0≤mk≤T0<h≤ho,and0<k≤ko. 我们引入记号 dh=(h∑ 所以上述稳定性等价于 ll≤c∑‖‖ 5
with the partial differential equation if for any smooth function φ(t, x) P φ − Pk,hφ → 0 as k, h → 0, the convergence being pointwise convergence at each grid point. Example The lax-Friedrichs Scheme éLax-Friedrichs ªµ Pk,hφ = φ n+1 m − 1 2 (φ n m+1 + φ n m−1 ) k + a φ n m+1 − φ n m−1 2h ^Taylorúªµ φ n m±1 = φ n m ± hφx + 1 2 h 2φxx ± 1 6 h 3φxxx + O(h 4 ) ·± Pk,hφ = φt + aφx + 1 2 kφtt − 1 2 k −1h 2φxx + 1 6 ah2φxxx + O(h 4 + k −1h 4 + k 2 ) ùh, k → 0§ k −1h 2 → 0§Pk,hφ − P φ → 0. Definition. 4 A finite difference scheme Pk,hv n m = 0 for a first-order equation is stable if there is an integer J and positive numbers h0 and k0 such that for any positive time T, there is a constant CT such that h X∞ m=−∞ |v n m| 2 ≤ CT h X J j=0 X∞ m=−∞ |v j m| 2 for 0 ≤ nk ≤ T,0 < h ≤ h0, and 0 < k ≤ k0. ·Ú\PÒ kwkh = h X∞ m=−∞ |vm| 2 !1 2 ¤±þã½5 du kv n kh ≤ C ∗ T X J j=0 kv j kh 5
Example:考虑计算格式 m+ B 这个计算格式当a|+|≤1时是稳定的。 差分方程的稳定性与PDE方程初值问题的适定性是密切相关的。 Definition. 5 The initial value problem for the first-order partial d ifferen- tial equation Pu=0 is well-posed if for any time T>0, there is a constant CT such that any solution u(t, a)safisfies 1,) d z CT/a(0,)dfor0≤t≤T 1.1.5 The Lax-Richtmyer Equivalence Theorem Theorem. 6 The Lat-Richtmyer Equivalence Theorem. A consistent finite difference scheme for a partial differential equation for which the initial value problem is well-posed is convergent if and only if it is stable 1.1.6 The Courant-Friedrichs-Lewy Condition Theorem. 7 For an explicit scheme for the hyperbolic equation +a-=0 of the form n+1 aum_1+ Bum+rr n+1 with k/h=a held constant, a necessary condition for stability is the Courant friedrichs-Lewy (CFL) condition, a<1 For systems of equations for which v is a vector and a, B and y are matrices, we must have aidl s 1 for all eigenvalues ai of the matriT A
Example: ÄOªµ v n+1 m = αvn m + β n m+1 ùOª|α| + |β| ≤ 1´½" ©§½5PDE§Ð¯K·½5´'" Definition. 5 The initial value problem for the first-order partial d ifferential equation P u = 0 is well-posed if for any time T ≥ 0, there is a constant CT such that any solution u(t, x) safisfies Z ∞ −∞ |u(t, x)| 2 dx ≤ CT Z ∞ −∞ |u(0, x)| 2 dx for 0 ≤ t ≤ T. 1.1.5 The Lax-Richtmyer Equivalence Theorem Theorem. 6 The Lax-Richtmyer Equivalence Theorem. A consistent finite difference scheme for a partial differential equation for which the initial value problem is well-posed is convergent if and only if it is stable. 1.1.6 The Courant-Friedrichs-Lewy Condition Theorem. 7 For an explicit scheme for the hyperbolic equation ∂u ∂t + a ∂u ∂x = 0 of the form v n+1 m = αvn m−1 + βvn m + γvn m+1 with k/h = λ held constant, a necessary condition for stability is the Courantfriedrichs-Lewy(CFL) condition, |aλ| < 1. For systems of equations for which v is a vector and α, β and γ are matrices, we must have |aiλ| ≤ 1 for all eigenvalues ai of the matrix A 6
Theorem. 8 There are no explicit, unconditionally stable, consistent finite difference schemes for hyperbolic systems of partial differential equations 这个结论发表在 Courant, R, K.O.Friedrichs, and H. Lewy, 1928, Uber die partiellen dif- ferenzengleichungen der mathematischen physik, Mathematische Annalen, 100: 32- 74
Theorem. 8 There are no explicit, unconditionally stable, consistent finite difference schemes for hyperbolic systems of partial differential equations. ù(ØuL3µ Courant, R., K.O.Friedrichs, and H.Lewy, 1928, Uber die partiellen dif- ¨ ferenzengleichungen der mathematischen physik, Mathematische Annalen,100:32- 74 7
Chapter 2 Analysis of Finite Difference Schemes 2.1 Fourier Analysis 当函数(x)定义在实轴R上,它的 Fourier变换(u)定义为 a(∞) Fourier反变换定义为 u(a) 类似地,如果网格函数定义在整数网格m∈Z上,它的 Fourier变化定义 为:对∈[-丌,]且(-丌)=0(丌) (5) Fourier反变换定义为
Chapter 2 Analysis of Finite Difference Schemes 2.1 Fourier Analysis ¼êu(x)½Â3¢¶Rþ,§FourierCuˆ(ω)½Â uˆ(ω) = 1 √ 2π Z ∞ −∞ u(x)e −iωxdx FourierC½Â u(x) = 1 √ 2π Z ∞ −∞ uˆ(ω)e iωxdω aq/§XJ ¼êv½Â3ê m ∈ Zþ,§FourierCz½Â µéξ ∈ [−π, π] vˆ(−π) = ˆv(π) vˆ(ξ) = 1 √ 2π X∞ m=−∞ vme −imξ FourierC½Â vm = 1 √ 2π Z π −π veˆ imξ(ξ)dξ 8
如果网格点之间的距离是h,通过变量代换,我们可以定义:对∈ h,r/] 6(E)=√2m=-∞ 反变换公式是 i(s)ende 个重要的结果是 Parseval关系:在连续情况下 Ja(a)ld.r a(u)2dx台‖l|2=‖al|2 类似的,在离散情况下 (5)2d vmPh=‖l2 Parseval等式广泛被用于稳定性分析。前面定义的稳定性估计可以改成等 价的形式: lh≤G∑0lh 2.2 Fourier Analysis and Partial Differential Equations 通过对 Fourier反变换公式求导,我们可以得到 e"wwwi(w)dw 因此我们有 D/)()=ia(u) 从这里可以看到,通过 Fourier变化,可以把求导变成乘法,这样使得我们 可以把PDE中的问题看成是代数问题 9
XJ :mål´h§ÏLCþ§·±½Âµéξ ∈ [−π/h, π/h], vˆ(ξ) = 1 √ 2π X∞ m=−∞ vme −imhξh Cúª´ vm = 1 √ 2π Z π/h −π/h vˆ(ξ)e imhξdξ (J´Parseval'Xµ3ëY¹e Z ∞ −∞ |u(x)| 2 dx = Z ∞ −∞ |uˆ(ω)| 2 dx ⇔ kuk2 = kuˆk2 aq§3lѹe kvˆk 2 h = Z π/h −π/h |vˆ(ξ)| 2 dξ = X∞ m=−∞ |vm| 2h = kvk 2 h Parseval ª2^u½5©Û"c¡½Â½5O±U¤ d/ªµ kvˆ n kh ≤ C ∗ T X J j=0 kvˆ j kh 2.2 Fourier Analysis and Partial Differential Equations ÏLéFourierCúª¦§·± ∂u ∂x(x) = 1 √ 2π Z ∞ −∞ e iωxiωuˆ(ω)dω Ïd·k ˆ∂u ∂x! (ω) = iωuˆ(ω) lùp±w§ÏLFourierCz§±r¦C¤¦{§ù¦· ±rPDE¥¯Kw¤´ê¯K" 9